Metamath Proof Explorer


Theorem adddird

Description: Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses addcld.1
|- ( ph -> A e. CC )
addcld.2
|- ( ph -> B e. CC )
addassd.3
|- ( ph -> C e. CC )
Assertion adddird
|- ( ph -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )

Proof

Step Hyp Ref Expression
1 addcld.1
 |-  ( ph -> A e. CC )
2 addcld.2
 |-  ( ph -> B e. CC )
3 addassd.3
 |-  ( ph -> C e. CC )
4 adddir
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )
5 1 2 3 4 syl3anc
 |-  ( ph -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )